Bag $I$ contains $3$ red,$4$ black,and $3$ white balls. Bag $II$ contains $2$ red,$5$ black,and $2$ white balls. One ball is transferred from Bag $I$ to Bag $II$,and then a ball is drawn from Bag $II$. The ball drawn is found to be black. What is the probability that the transferred ball was red?

$A$ company produces $10,000$ items per day. On a particular day,$2500$ items were produced on machine $A$,$3500$ on machine $B$,and $4000$ on machine $C$. The probability that an item produced by machines $A, B, C$ is defective is $2 \%$,$3 \%$,and $5 \%$,respectively. If one item is selected at random from the output and is found to be defective,then the probability that it was produced by machine $C$ is:

$A$ bag contains $4$ red and $4$ black balls,another bag contains $2$ red and $6$ black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Bag $A$ contains $4$ green and $3$ red balls and bag $B$ contains $4$ red and $3$ green balls. One bag is selected at random and a ball is drawn,which is found to be green. What is the probability that it was drawn from bag $B$?

Bag $A$ contains $2$ white and $3$ red balls and bag $B$ contains $4$ white and $5$ red balls. If one ball is drawn at random from one of the bags and is found to be red,then the probability that it was drawn from the bag $B$ is

There are $10$ coins in a box,out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time,then the probability that the selected coin has heads on both sides is